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If you're using Mint, search for the "Pulseaudio Device Chooser" in your Package Manager. It allows you to configure your sound server so the sound goes through the HDMI cable. Once the cable is plugged in, look at the Volume Control function for Output devices and select the one for HDMI. If there is no HDMI listed in the choices, try choosing one of the Digital Stereo selections, that's how I got mine to work.

Go back into the PulseAudio Manager and make sure all the volume sliders are up, and see if there are any other configuration bits that need to be enabled. HDMI should work easily in Linux Mint.

If you still can't figure out what the problem is, it might help to tell us more info, like what version of Linux Mint, what kind of hardware you are running it on, what kind of video card and are you using the built in sound hardware, or do you have a sound card like a SoundBlaster. There are no generic answers other than the hints I've already given you.